Evaluate Using L'Hospital's Rule limit as x approaches 8 of arctan(e^x)

Problem

(lim_x→8)(arctan(ex))

Solution

1. Identify the type of limit and whether L'Hospital's Rule is applicable.

2. Evaluate the limit by direct substitution, as the function ƒ(x)=arctan(ex) is continuous at x=8

3. Determine if an indeterminate form like 0/0 or ∞/∞ exists. Since substituting x=8 results in a finite value, L'Hospital's Rule is not required and cannot be applied.

4. Substitute the value x=8 into the expression to find the limit.

Final Answer

(lim_x→8)(arctan(ex))=arctan(e8)

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